I have one basic following function.
$$
L(y,t)=\begin{cases}
(1-\tau)(y-t)^2 &\text{ if } y<t, \\
\tau (y-t)^2 &\text{ if } y\geq t
\end{cases}
$$
where $y \in \mathbb{R}. \tau \in (0,1)$. Is that function is Lipschitz Continuous? If yes,then how it can be proved?
I have tried in the following way. By definition
\begin{align}
&|L(y,t_1)-L(y,t_2)|\\
=&|(1-\tau)(y-t_1)^2+\tau(y-t_1)^2-(1-\tau)(y-t_2)^2-\tau(y-t_2)^2|\\
=&\left|(1-\tau)\left[(y-t_1)^2-(y-t_2)^2\right]
+\tau\left[(y-t_1)^2-(y-t_2)^2\right]\right|
\end{align}
I am stuck here..because it involves the terms of $y|y <t_1$ and $y|y\geq t_2$ and similarly for $< t_2$ and $\geq t_2$